Problem: $ C = \left[\begin{array}{rrr}3 & 5 & 3 \\ -2 & -1 & -2\end{array}\right]$ $ B = \left[\begin{array}{rr}-1 & 2 \\ 5 & 2 \\ -1 & -1\end{array}\right]$ What is $ C B$ ?
Because $ C$ has dimensions $(2\times3)$ and $ B$ has dimensions $(3\times2)$ , the answer matrix will have dimensions $(2\times2)$ $ C B = \left[\begin{array}{rrr}{3} & {5} & {3} \\ {-2} & {-1} & {-2}\end{array}\right] \left[\begin{array}{rr}{-1} & \color{#DF0030}{2} \\ {5} & \color{#DF0030}{2} \\ {-1} & \color{#DF0030}{-1}\end{array}\right] = \left[\begin{array}{rr}? & ? \\ ? & ?\end{array}\right] $ To find the element at any row $i$ , column $j$ of the answer matrix, multiply the elements in row $i$ of the first matrix, $ C$ , with the corresponding elements in column $j$ of the second matrix, $ B$ , and add the products together. So, to find the element at row 1, column 1 of the answer matrix, multiply the first element in ${\text{row }1}$ of $ C$ with the first element in ${\text{column }1}$ of $ B$ , then multiply the second element in ${\text{row }1}$ of $ C$ with the second element in ${\text{column }1}$ of $ B$ , and so on. Add the products together. $ \left[\begin{array}{rr}{3}\cdot{-1}+{5}\cdot{5}+{3}\cdot{-1} & ? \\ ? & ?\end{array}\right] $ Likewise, to find the element at row 2, column 1 of the answer matrix, multiply the elements in ${\text{row }2}$ of $ C$ with the corresponding elements in ${\text{column }1}$ of $ B$ and add the products together. $ \left[\begin{array}{rr}{3}\cdot{-1}+{5}\cdot{5}+{3}\cdot{-1} & ? \\ {-2}\cdot{-1}+{-1}\cdot{5}+{-2}\cdot{-1} & ?\end{array}\right] $ Likewise, to find the element at row 1, column 2 of the answer matrix, multiply the elements in ${\text{row }1}$ of $ C$ with the corresponding elements in $\color{#DF0030}{\text{column }2}$ of $ B$ and add the products together. $ \left[\begin{array}{rr}{3}\cdot{-1}+{5}\cdot{5}+{3}\cdot{-1} & {3}\cdot\color{#DF0030}{2}+{5}\cdot\color{#DF0030}{2}+{3}\cdot\color{#DF0030}{-1} \\ {-2}\cdot{-1}+{-1}\cdot{5}+{-2}\cdot{-1} & ?\end{array}\right] $ Fill out the rest: $ \left[\begin{array}{rr}{3}\cdot{-1}+{5}\cdot{5}+{3}\cdot{-1} & {3}\cdot\color{#DF0030}{2}+{5}\cdot\color{#DF0030}{2}+{3}\cdot\color{#DF0030}{-1} \\ {-2}\cdot{-1}+{-1}\cdot{5}+{-2}\cdot{-1} & {-2}\cdot\color{#DF0030}{2}+{-1}\cdot\color{#DF0030}{2}+{-2}\cdot\color{#DF0030}{-1}\end{array}\right] $ After simplifying, we end up with: $ \left[\begin{array}{rr}19 & 13 \\ -1 & -4\end{array}\right] $